N-th Tribonacci Number

Derive directly from the recurrence

With the standard definition T0 = 0, T1 = 1, T2 = 1, and Tn = T(n-1) + T(n-2) + T(n-3), return immediately for n < 3. Otherwise keep three variables representing consecutive previous values. For each index from three through n, compute next, shift the window, and continue.

The invariant is that before an iteration, the variables equal T(i-3), T(i-2), and T(i-1); after shifting they represent the next triple. This uses O(n) time and O(1) auxiliary space. A DP array is also correct but unnecessary when only the final value is needed. Discuss integer range if constraints can make values exceed the language's integer type. Test n = 0, 1, and 2 separately, then 3 to catch an off-by-one error. If callers need all terms, preserve the array and state O(n) space honestly.